In this paper we attempt a self-contained approach to infinite dimensional
Hamiltonian systems appearing from holomorphic curve counting in Gromov-Witten
theory. It consists of two parts. The first one is basically a survey of Dubrovin’s
approach to bihamiltonian tau-symmetric systems and their relation with Frobenius
manifolds. We will mainly focus on the dispersionless case, with just some hints
on Dubrovin’s reconstruction of the dispersive tail. The second part deals with the
relation of such systems to rational Gromov-Witten and Symplectic Field Theory.
We will use Symplectic Field theory of S1×M as a language for the Gromov-Witten
theory of a closed symplectic manifold M. Such language is more natural from the
integrable systems viewpoint. We will show how the integrable system arising from
Symplectic Field Theory of S1×M coincides with the one associated to the Frobenius
structure of the quantum cohomology of M.